39 research outputs found
Maximal planar scale-free Sierpinski networks with small-world effect and power-law strength-degree correlation
Many real networks share three generic properties: they are scale-free,
display a small-world effect, and show a power-law strength-degree correlation.
In this paper, we propose a type of deterministically growing networks called
Sierpinski networks, which are induced by the famous Sierpinski fractals and
constructed in a simple iterative way. We derive analytical expressions for
degree distribution, strength distribution, clustering coefficient, and
strength-degree correlation, which agree well with the characterizations of
various real-life networks. Moreover, we show that the introduced Sierpinski
networks are maximal planar graphs.Comment: 6 pages, 5 figures, accepted by EP
Statistical Mechanics of Two-dimensional Foams
The methods of statistical mechanics are applied to two-dimensional foams
under macroscopic agitation. A new variable -- the total cell curvature -- is
introduced, which plays the role of energy in conventional statistical
thermodynamics. The probability distribution of the number of sides for a cell
of given area is derived. This expression allows to correlate the distribution
of sides ("topological disorder") to the distribution of sizes ("geometrical
disorder") in a foam. The model predictions agree well with available
experimental data
A Topological Glass
We propose and study a model with glassy behavior. The state space of the
model is given by all triangulations of a sphere with nodes, half of which
are red and half are blue. Red nodes want to have 5 neighbors while blue ones
want 7. Energies of nodes with different numbers of neighbors are supposed to
be positive. The dynamics is that of flipping the diagonal of two adjacent
triangles, with a temperature dependent probability. We show that this system
has an approach to a steady state which is exponentially slow, and show that
the stationary state is unordered. We also study the local energy landscape and
show that it has the hierarchical structure known from spin glasses. Finally,
we show that the evolution can be described as that of a rarefied gas with
spontaneous generation of particles and annihilating collisions
Manipulation and removal of defects in spontaneous optical patterns
Defects play an important role in a number of fields dealing with ordered
structures. They are often described in terms of their topology, mutual
interaction and their statistical characteristics. We demonstrate theoretically
and experimentally the possibility of an active manipulation and removal of
defects. We focus on the spontaneous formation of two-dimensional spatial
structures in a nonlinear optical system, a liquid crystal light valve under
single optical feedback. With increasing distance from threshold, the
spontaneously formed hexagonal pattern becomes disordered and contains several
defects. A scheme based on Fourier filtering allows us to remove defects and to
restore spatial order. Starting without control, the controlled area is
progressively expanded, such that defects are swept out of the active area.Comment: 4 pages, 4 figure
The Harris-Luck criterion for random lattices
The Harris-Luck criterion judges the relevance of (potentially) spatially
correlated, quenched disorder induced by, e.g., random bonds, randomly diluted
sites or a quasi-periodicity of the lattice, for altering the critical behavior
of a coupled matter system. We investigate the applicability of this type of
criterion to the case of spin variables coupled to random lattices. Their
aptitude to alter critical behavior depends on the degree of spatial
correlations present, which is quantified by a wandering exponent. We consider
the cases of Poissonian random graphs resulting from the Voronoi-Delaunay
construction and of planar, ``fat'' Feynman diagrams and precisely
determine their wandering exponents. The resulting predictions are compared to
various exact and numerical results for the Potts model coupled to these
quenched ensembles of random graphs.Comment: 13 pages, 9 figures, 2 tables, REVTeX 4. Version as published, one
figure added for clarification, minor re-wordings and typo cleanu
Structure and dynamics of cellular systems
This review addresses recent progress in the analysis and modelling of disordered cell structures. It focuses on planar systems, for which most research has been performed. The subject is approached from a general viewpoint rather than focusing on the specific evolution of certain systems. To this end, results of studies performed in completely different disciplines, ranging from applied sciences, biology and physics to mathematics, are gathered and discussed systematically. Special emphasis is laid on common properties of the different structures, including a critical discussion of the information contained in typically measured quantities. Efficient techniques for the simulation of typical disordered cell structures are summarized, and novel theoretical approaches to the modelling of dynamical cell structures are presented. Here, special attention is paid to the application of the methods of statistical mechanics and the resulting implications. In several systems, a breakdown of order induced by variation of an external control parameter can be observed. Its diagnostics in Voronoi tessellations is presented focusing on the connection with the underlying physics. The discussion of planar cell structures is concluded with a presentation of the peculiar properties of exceptional systems, characterized by the coexistence of two different length scales or by scale invariance. This review concludes with a presentation of recent developments in the research on three-dimensional cell structures